bathroom and washroom remodel

Hi,
I'm remodeling our bathroom and moving a washer and dryer to the room behind the bath. I've been reading through the posts, which have been very helpful and have a diagram I would love to get some feedback on. I am using some of the existing plumbing which was done a few years ago. I'm adding the washer, moving the WC and adding a utility sink.

A friend suggested that I connect the washer below the toilet to avoid bubbles. That is the reason for the parallel 3" line. Is that necessary? Also, do I have enough venting?

Are there vents at the lava nd the utility sink? I guess under some conditions you could call what you have a circuit vent but that isn't used too often in residential construction and your vents would be too small for a properly sized circuit.

The utility lav would use an AAV too. Is it possible to attach the washer before the toilet or is it best to attach it after the toilet? I can add non-AAV venting to the washer and the toilet without too much work. How close to the toilet do I need to have the vent? Can it not come out of the main line?

for all practical purposes, you do NOT have ANY vents, at least none where they would perform their intended purposes. You have also created a "wasteful" system as far as the cost of installing it is concerned. DON'T listen to friends who do not know what they are talking about. You do not need that 3" line to the washing machine. IF you vent the system properly you will not get "gurgling". IF you do not vent it properly, it will gurgle regardless of where you connect it or what size pipe you use.